We would normally refer to that as September 11 because it's much more talked about in the US, and that's the phrase used there.

Any other dates will likely be in the same order as written. For instance, the rhyme for bonfire night is 'remember remember, the fifth of November'. I believe that many in the US also talk about the fourth of July, rather than July fourth, so it's not like English has the hard-and-fast rule you were proposing.

Ok, fair enough. What do you say for non-special dates like July 3rd, 2015?

'Third of July, 2015' or more likely 'third of July'. The date format really isn't lying to us in UK English.

As a British English speaker, I'd say "yes".

Technically, I'd drop the "th of" and just say/write "11 September 2001".

Interesting. Would you choose "three July twenty fifteen", "July third twenty fifteen", or "the third of July twenty fifteen" (substitute "two thousand" for "twenty" if you like)? Assume someone has asked you the date and you're responding out loud.

I would say "three July twenty fifteen".

It took me a while to figure this out because actually it's quite rare to speak a date including a year without reading it - most spontaneously-spoken dates are this year (so the year is implied) and for a read dates, I'd probably say whatever was written.

The clincher was how I'd say my birth date, which would be of the form above.

I'm not claiming to be the definitive British English speaker, though! ;)

...and as another poster commented, it might depend on context - for example, "September 11" is often used in British English because it refers to an American event.

As a Brit, IMO both are perfectly acceptable in English prose. It isn't unusual to say "October the 4th" as opposed to "the 4th of October".